Modulation Transfer Function (MTF)
(This method works at any magnification, but involves building a target, as described below.)
Fig 1. Loss of contrast
Ideally, light from the subject would map directly to the image without any spreading or distortion. But lens imperfections (eg., aberrations) and aperture diffraction spread light from the subject. The spreading decreases contrast, darkening neighbouring areas that should have been brighter, and brightening adjacent areas that should have been darker. Eventually, where neighbouring black and white areas on the subject are very close, the corresponding image points are just grey. This is illustrated in figure 1, where lines of various spatial wavelengths have been treated with a Gaussian blur of radius 1.5 pixels -- but note also that even the long wavelengths, though still quite visible, lost clarity and sharpness as their contrast was diminished by blurring.
The quality of a (stereo)microscope can be measured by how well it transmits spatial frequencies; higher quality microscopes will transmit higher spatial frequencies. The 'Modulation Transfer Function' (MTF) is a measure of the ability of an optical system to transfer contrast from the subject to the image plane, as a function of spatial frequency (so it's a spatial frequency response curve).
Fig 2. Five degree slant
One way to measure the MTF of an optical system involving digital sensors is to analyse how well an edge is imaged; the better the optical system, the sharper the edge in the image. By arranging the edge be slanted slight with respect to the sensor pixel rows, the edge response in the image can be measured with high resolution (clever!), as described in the documentation pages for two software packages that produce MTF reports from given images of slanted edges: Imatest and QuickMTF.
The test image must have a straight edge and must be uniform intensity on either side, one side darker than the other. To avoid saturation, neither side should be white or black (two shades of grey are best). The contrast between the two sides should be low enough not to trigger the sharpening algorithms of cameras that cannot provide Raw data.
Fig 3. Razor blade target, above grey surface
It's essential that the edge be straight, and it's tough finding or making an object or drawing with perfectly straight edge at microscopic scales. I used a razor blade. To make it uniformly coloured and dark, I immersed one side briefly in a candle flame to deposit a carbon coating, just enough to make it uniformly black, taking care not to create lumps (it's easy to wipe off a deposit and try again). Then the blade was mounted a few millimeters above a uniform grey surface (a photographic print fragment, in my case); the millimeters gap ensured that the underlying surface would be out of focus and thus more uniform. The resulting target, shown in figure 3, was then illuminated from a side so that there would be no shadow visible when the edge is viewed from the top by the microscope.
The edge was then photomicrographed through a stereomicroscope at various zoom levels. It's crucial that the edge be in focus, and slanted close to 5 degrees with respect to the camera's pixel rows (or columns), so that the resulting photomicrograph looks like figure 2. To obtain the desired angle, I used a wedge of paper cut at 5 degrees as a reference against the frame of LCD display of the camera. The edge can be positioned in the frame wherever you like, but best optical performance is likely to be on the optical axis. Illumination can be adjusted to control contrast.
To ensure that at least a portion of the edge passes through the focal plane, perfectly in focus, the target can be propped up slightly on one end (or rely upon the angled view of a single stereomicroscope objective).
As mentioned, I'm aware of two image analyser packages that provide MTF analysis: Imatest, and QuickMTF. Both are commercial products but offer free evaluation trials. Imatest is used by a number of photographic review sites (eg., dpReview.com), QuickMTF is simpler and easier to use. They differ slightly in the results; Imatest tends to report better performance than QuickMTF. Below are examples of their analyses of the same image (which was photograph using a Canon A75 of a low-contrast target generated by an LCD display).
Fig 4. Imatest (edge profile and MTF on one report page)
Fig 5. QuickMTF (edge profile or MTF can be displayed, but not same page)
A single value for resolution or numerical aperture is somewhat less interesting when one has an MTF curve, but the Rayleigh resolution and equivalent diffraction-limited NA at various MTF levels can be estimated from MTF figures provided by the image analyser software as follows:
The diffraction-limited MTF function for a uniformly lit circular aperture is:
MTF(ν) = 2/π (φ - cos(φ) sin(φ)),
where ν is spatial frequency (cycles/mm) and φ = arccos(ν λ / 2NA)
MTF is zero at the 'cut-off' frequency where φ is 0, ie., ν = 2NA / λ. MTF50 is obtained when φ = 1.155, therefore:
1.155 = arccos(νMTF50 λ / 2NA)
cos(1.155) = νMTF50 λ / 2NA
NA = νMTF50 λ / 0.808
NA = νMTF30 λ / 1.17
According to the Rayleigh Criterion, two points in the subject plane are considered resolvable when their Airy disc centers are separated by at least the radius of an Airy disc. The radius rAiry of the Airy disc (the distance from the central point to the first minimum ring) is:
rAiry = 0.61 λ / NA
Thus the Rayleigh frequency νRayleigh is:
νRayleigh = 1 / rAiry = NA / 0.61 λ
We can find the NA from the MTF at the Rayleigh frequency, as follows:
φRayleigh = arccos(νRayleigh λ / 2NA) = arccos((NA / 0.61 λ)(λ / 2NA)) = arccos(1 / 1.22)
MTF(νRayleigh) = 0.0894 ≅ 9%
rRayleigh = 1 / νMTF09
And with the same procedure used to find MTF50 above,
NA = νMTF09 λ / 1.638
However, the above estimates of NA from MTF assume that the optics are diffraction-limited. If there are optical errors (eg., aberrations), the MTF curve will be lower than what it would be were the optics diffraction-limited, and thus the NA will be under-estimated, and likely will vary depending on which MTF value is used (eg., MTF50, MTF30 or MTF09). NA depends on geometry, but that doesn't ensure the optics can deliver. MTF is a measure of complete system performance, including the NA and optics. NA as derived from MTF as above could be considered an 'effective NA', ie., the NA of a system with diffraction-limited optics that would deliver the MTF observed. Given this, probably more useful is the Rayleigh resolution (or the equiv. frequency) based on MTF09.
See Review: MicroscopeNet V434B stereomicroscope for an example of the application of this method applied to a microscope.